Definition 4.11.2 (Quotient Map). Let $X, Y$ be topological spaces and $\pi: X \to Y$ be a surjective map, then the following are equivalent:
For any $U \subset Y$, $U$ is open if and only if $\pi^{-1}(U)$ is open.
$\pi \in C(X; Y)$, and for any $U \subset X$ saturated and open, $\pi(U)$ is open.
If the above holds, then $\pi$ is a quotient map.
Proof. $(1) \Rightarrow (2)$: Let $U \subset Y$ be open, then $\pi^{-1}(U)$ is open, so $f \in C(X; Y)$. If $V \subset X$ is saturated and open, then $\pi(V) = \pi(\pi^{-1}(\pi(V)))$ is open.
$(2) \Rightarrow (1)$: Let $U \subset Y$ be open, then $\pi^{-1}(U)$ is open by continuity. If $V \subset Y$ and $\pi^{-1}(V)$ is open, then $V = \pi(\pi^{-1}(V))$ is open.$\square$